Theorem prnz 4496

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4484	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4122	1 ⊢ {𝐴, 𝐵} ≠ ∅

Syntax hints:   ∈ wcel 2157   ≠ wne 2969  Vcvv 3383  ∅c0 4113  {cpr 4368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-v 3385  df-dif 3770  df-un 3772  df-nul 4114  df-sn 4367  df-pr 4369

This theorem is referenced by:  opnz  5130  propssopi  5162  fiint  8477  wilthlem2  25144  upgrbi  26320  wlkvtxiedg  26866  shincli  28738  chincli  28836  spr0nelg  42513  sprvalpwn0  42520